in this paper, we verify the solvability of free product of finite cyclic groups with topological methods. we use cayley graphs and everitt methods to construct suitable 2-complexes corresponding to the presentations of groups and their commutator subgroups. in particular, using these methods, we prove that the commutator subgroup of $mathbb{z}_{m}*mathbb{z}_{n}$ is free of rank $(m-1)(n-1)$ fo...

One of the most important problems of fuzzy group theory is to classify the fuzzy subgroups of a finite group. This topic has enjoyed a rapid development in the last few years. Several papers have treated the particular case of finite abelian groups. Thus, in [13] the number of distinct fuzzy subgroups of a finite cyclic group of square-free order is determined, while [14–16] and [28] deal with...

let $g$ be a finite group with the identity $e$‎. ‎the subgroup intersection graph $gamma_{si}(g)$ of $g$ is the graph with vertex set $v(gamma_{si}(g)) = g-e$ and two distinct vertices $x$ and $y$ are adjacent in $gamma_{si}(g)$ if and only if $|leftlangle xrightrangle capleftlangle yrightrangle|>1$‎, ‎where $leftlangle xrightrangle $ is the cyclic subgroup of $g$ generated by $xin g$‎. ‎in th...

We associate a graph ΓG to a non locally cyclic group G (called the non-cyclic graph of G) as follows: take G\Cyc(G) as vertex set, where Cyc(G) = {x ∈ G | 〈x, y〉 is cyclic for all y ∈ G}, and join two vertices if they do not generate a cyclic subgroup. We study the properties of this graph and we establish some graph theoretical properties (such as regularity) of this graph in terms of the gro...

let $g$ be a group and $x in g$‎. ‎the cyclicizer of $x$ is defined to be the subset $cyc(x)=lbrace y in g mid langle x‎, ‎yrangle ; {rm is ; cyclic} rbrace$‎. ‎$g$ is said to be a tidy group if $cyc(x)$ is a subgroup for all $x in g$‎. ‎we call $g$ to be a c-tidy group if $cyc(x)$ is a cyclic subgroup for all $x in g setminus k(g)$‎, ‎where $k(g)$ is the intersection of all the cyclicizers in ...

‎let $g$ be a finite group which is not a cyclic $p$-group‎, ‎$p$ a prime number‎. ‎we define an undirected simple graph $delta(g)$ whose‎ ‎vertices are the proper subgroups of $g$, which are not contained in the‎ ‎frattini subgroup of $g$ and two vertices $h$ and $k$ are joined by an edge‎ ‎if and only if $g=langle h‎ , ‎krangle$‎. ‎in this paper we classify finite groups with planar graph‎. ‎...

Following Malcev [5], we will call a subgroup M of a group G finitely separable if for any element g ∈ G, not belonging to M, there exists a homomorphism φ of group G onto some finite group X such that gφ / ∈Mφ. This is equivalent to the statement that for any element g ∈ G \M, there exists a normal subgroup N of finite index in G such that g / ∈MN . A group G is subgroup separable if each of i...

We give a characterization for fundamental groups of graphs of groups amalgamating cyclic edge subgroups to be cyclic subgroup separable if each pair of edge subgroups has a non-trivial intersection. We show that fundamental groups of graphs of abelian groups amalgamating cyclic edge subgroups are cyclic subgroup separable, hence residually finite, if each edge subgroup is isolated in its conta...